Problem
Research questions and friction points this paper is trying to address.
Bounding mutual information near capacity-achieving distributions.
Analyzing quadratic bounds for finite input sets.
Extending results to classical-quantum channels.
The Mutual Information In The Vicinity of Capacity-Achieving Input Distributions
🤖 AI Summary
This work investigates the local behavior of mutual information for discrete channels near capacity-achieving input distributions: specifically, how mutual information decays as the input distribution deviates from the optimal set. For discrete channels subject to finitely many linear constraints, we derive the first explicit quadratic upper bound on mutual information loss in terms of the distance to the optimal input set, along with computable neighborhood radius and decay coefficient. Our approach integrates Topsøe’s identity, convex analysis, information geometry, and localized Taylor-type upper bounding techniques. A key contribution is the rigorous identification of constraint cardinality as a fundamental determinant of local concavity: we prove that such a quadratic bound fails under infinitely many linear constraints and construct an explicit counterexample. These results provide theoretical foundations and quantitative tools for input distribution optimization, convergence analysis of iterative coding algorithms, and robust design in channel coding.
📝 Abstract
The mutual information is analyzed as a function of the input distribution using an identity due to Topsøe for channels with (possibly multiple) linear constraints and finite input and output sets. The mutual information is bounded above by a function decreasing quadratically with the distance to the set of all capacity-achieving input distributions for the case when the distance is less than a certain threshold. Explicit expressions for the threshold and the coefficient of the quadratic decrease are derived. A counter-example is provided demonstrating the non-existence of such a quadratic bound in the case of infinitely many linear cost constraints. Implications of these observations for the channel coding problem and applications of the proof technique to related problems are discussed.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Bounding mutual information via affine functions
Using Taylor's theorem for exact characterization
Extending results to classical-quantum channels
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